Weak Hamburger-type weighted shifts and their examples

Let Wα be a bounded weighted shift with weight sequence α={αn}n=0∞ and let γn=α02⋯αn−12(n≥1) with γ0=1. The positivity of both of the infinite matrices (γi+j)0≤i,j<∞ and (γi+j+1)0≤i,j<∞ is a condition equivalent to subnormality of Wα. For a positive integer n, the positivity of (γi+j)0≤i,j